Harry Redknapp's men have stormed into the top three of the Premier League following a run of just one defeat from 19 matches.
Spurs head to the Etihad Stadium on Sunday five points behind leaders Manchester City following last weekend's 1-1 draw at home to Wolves.
Redknapp has urged his side to believe they can still have a say in the outcome of the championship, although admitted it will be a tough ask to overhaul both City and Manchester United - who visit White Hart Lane in March.
Centre-back King feels everything has to be kept in perspective.
"We are in a great position at the moment and have probably surprised a few people," King told talkSPORT.
"At this stage of the season, to be talked about as still capable of winning the title is great for us.
"We are not getting too carried away at the moment, though. We are just getting our heads down and grinding out results."
He added: "We look at the league table and we can see we are in amongst it.
"That is new to us and maybe people would expect us to fall away in the second half of the season but, as players, we believe in our ability and we believe we can keep getting results.
"We have lost just one in the last 19 and that is great form. If we keep it up then we are in with a chance."
Redknapp remains open to doing business in the January transfer window if he can find the right player.
Spurs, however, still have plenty of other options in reserve, with the likes of Heurelho Gomes, Niko Kranjcar, Roman Pavlyuchenko, Steven Pienaar and Jermain Defoe all desperate for more game time.
Mexico midfielder Giovani dos Santos is said to be a target for Spanish club Villarreal.
